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Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Numere complexe, Cuaternioni si Aplicatii
Andrei Marcus
http://math.ubbcluj.ro/˜marcus/
Junior Summer University, 19 iulie 2011
Universitatea Babes-Bolyai Cluj-NapocaFacultatea de Matematica si Informatica
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Cuprins I
1 Numere complexeDefinitieIstoricPlanul complexRotatii ın plan
2 CuaternioniDefinitie si proprietatiRotatie ın jurul unei axe
3 Octonioni
4 Bibliografie
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Definitie
Numere complexe
x2 + 1 = 0; i2 = −1;
C = R× R
z = (a, b) = a + bi 7→(a −bb a
)(reprezentare matriceala)
conjugatul: z = a − bi
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Istoric
Numere complexe
Niccolo Tartaglia (1499–1557); Scipione del Ferro (1465–1526)Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Istoric
Numere complexe
Girolamo Cardano (1501–1576); Rene Descartes (1596–1650)Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Planul complex
Planul complex
Diagrama Argand: r =√zz , sin θ = y/r , cos θ = x/r .
Formula lui Euler: e iθ = cos θ + i sin θ; e iπ = −1.Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Planul complex
Planul complex
Caspar Wessel (1745–1818); Jean-Robert Argand (1768–1822)
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Planul complex
Euler (1707–1783)
Gauss (1777–1855)
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Planul complex
Carl Friedrich Gauss ın 1828
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Rotatii ın plan
Rotatia ın plan
(x ′
y ′
)=
(cos θ − sin θsin θ cos θ
)(xy
)
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Definitie si proprietati
Cuaternioni
Hamilton 1843: H = R× R× R× Rq = (a, b, c , d) = a + bi + cj + dk , undei2 = j2 = k2 = ijk = −1;
Gauss: ınmultirea cuadruplelor;
Lagrange: Teorema celor 4 patrate.
conjugatul: q := a − bi − cj − dk ,
norma: |q| =√qq;
q−1 = q/|q|2; q′ = q/|q| are norma 1.
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Definitie si proprietati
Cuaternioni
Sir William Rowan Hamilton (1805 – 1865)
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Definitie si proprietati
Cuaternioni
Reprezentare matriceala
q 7→(
z −ww z
)7→
a −b −c −db a −d cc d a −b−d c b a
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Definitie si proprietati
Cuaternioni: notatia vectoriala
q = s + ~v = [s, ~v ], unde s ∈ R, ~v = x~i + y~j + z~k
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Definitie si proprietati
Cuaternioni: notatia vectoriala
Fie
qa = sa + ~va = [sa, ~va] = sa + xa~i + ya~j + za~k ,
qa = sb + ~vb = [sb, ~vb] = sb + xb~i + yb~j + zb~k .
Avem
qaqb = (sasb −~a · ~b) + sa~b + sb~a +~a × ~b,
unde
~va · ~vb = xaxb + yayb + zazb;
~va × ~vb =
∣∣∣∣∣∣~i ~j ~kxa ya zaxb yb zb
∣∣∣∣∣∣Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Rotatie ın jurul unei axe
Rotatie ın jurul unei axe
Rotatie ın jurul unei axe; Olinde Rodrigues (1794–1851)
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Rotatie ın jurul unei axe
Rotatie ın jurul unei axe
Fie q = s + λ~v un cuaternion de lungime 1,adica |~v | = 1, s2 + λ2 = 2.Notam λ = sin θ si s = cos θ.Fie p = [0, ~p] un cuaternion pur. Calculam conjugatul lui p:
qpq−1 = [s, λ~v ][0, ~p][s,−λ~v ]
= [−λ~v · ~p, s~p + λ~v × ~p][s,−λ~v ]
= [−λs~v · p + λs~p · ~v + λ2(~v × ~p) · ~v ,λ2(~v · ~p)~v + s2~p + λs~v × ~p − λs~p × ~v − λ2(~v × ~p)× ~v ]
= [λ2(~v × ~p) · ~v , λ2(~v · ~p)~v + s2~p + 2λs~v × ~p − λ2(~v × ~p)× ~v ].
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Rotatie ın jurul unei axe
Rotatie ın jurul unei axe
Avem(~v × ~p) · ~v = 0
si(~v × ~p)× ~v = (~v · ~v)~p − (~p · ~v)~v = ~p − (~p · ~v)~v ,
deci
qpq−1 = [0, λ2(~v · ~p)~v + s2p + 2λs~v × ~p − λ2~p + λ2(~p · ~v)~v ]
= [0, 2λ2(~v · ~p)~v + (s2 − λ2)~p + 2λs~v × ~p].
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Rotatie ın jurul unei axe
Rotatie ın jurul unei axe
Inlocuind s = cos θ si λ = sin θ obtinem:
qpq−1 = [0, 2 sin2 θ(~v · ~p)~v + (cos2 θ − sin2 θ)~p + 2 sin θ cos θ~v × ~p]
= [0, (1− cos 2θ)(~v · ~p)~v + cos 2θ~p + sin 2θ~v × ~p].
Observam ca luand q = [cos 12θ, sin 1
2θ~v ],
obtinem exact formula lui Rodrigues:
qpq−1 = [0, (1− cos θ)(~v · ~p)~v + cos θ~p + sin θ~v × ~p],
deci p′ = qpq−1 este chiar rotatia de unghi θ a vectorului ~p ınjurul vectorului ~v .
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Rotatie ın jurul unei axe
Rotatie – forma matriceala
Fieq = s + ~v ;~v = x~i + y~j + z~k ;|q|2 = s2 + |~v |2 = s2 + x2 + y 2 + z2 = 1;
~p = xp~i + yp~j + zp~k ;
Atuncip′ = qpq−1 =2(s2 + x2)− 1 2(xy − sz) 2(xz + sy)
2(xy + sz) 2(s2 + y 2)− 1 2(yz − sx)2(xz − sy) 2(yz + sx) 2(s2 + z2)− 1
xpypzp
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Octonioni
O = H×H;
Definition
(p, q)(r , s) = (pr − sq, sp + qr)
x = (p, q) = x0e0 +x1e1 +x2e2 +x3e3 +x4e4 +x5e5 +x6e6 +x7e7e0 e1 e2 e3 e4 e5 e6 e7e1 −1 e3 −e2 e5 −e4 −e7 e6e2 −e3 −1 e1 e6 e7 −e4 −e5e3 e2 −e1 −1 e7 −e6 e5 −e4e4 −e5 −e6 −e7 −1 e1 e2 e3e5 e4 −e7 e6 −e1 −1 −e3 e2e6 e7 e4 −e5 −e2 e3 −1 −e1e7 −e6 e5 e4 −e3 −e2 e1 −1
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Octonioni
John Thomas Graves (1806–1870); Arthur Cayley (1821–1895)Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Octonioni
John Baez
The real numbers are the dependable breadwinner of thefamily, the complete ordered field we all rely on. The complexnumbers are a slightly flashier but still respectable youngerbrother: not ordered, but algebraically complete. Thequaternions, being noncommutative, are the eccentric cousinwho is shunned at important family gatherings. But theoctonions are the crazy old uncle nobody lets out of the attic:they are nonassociative.
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni
Cuprins Numere complexe Cuaternioni Octonioni Bibliografie
Bibliografie I
John Vince: Quaternions for Computer Graphics,Springer-Verlag, New York 2011.
http://www-history.mcs.st-and.ac.uk/history/
http://en.wikipedia.org/
Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni